First Year Engineering (Semester 1)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1(a)
Separate into a real part and imaginary part of $\cos^{-1} (\frac{3i}{4})$
(3 marks)
00
1(b)
Show that the matrix A is unitary where A = $\matrix{\alpha+i\gamma & -\beta+i\delta\\ \beta+i\delta & \alpha-i\gamma}$ is unitary if $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = 1$
(3 marks)
00
1(c)
If $z = \tan (y+ax)+(y-ax)^{\frac{3}{2}}$ then show that $\frac{\delta^2z}{\delta x^2} = a^2 \frac{\delta^2z}{\delta y^2}$
(3 marks)
00
1(d)
If $x = uv , y=\frac{u}{v}$ Prove that $JJ^/ = 1$
(3 marks)
00
1(e)
Find the $n^{th}$ derivative of $\frac{x^3}{(x+1)(x-2)}$
(4 marks)
00
1(e)
Using the matrix $A = \matrix{-1 & 2\\-1 &1}$decode the message matrix $ C = \matrix{4 & 11 & 12 & -2 \\ -4 & 4 & 9 & -2 }$
(4 marks)
00
2(a)
If $\sin^4\theta \cos^3\theta = a\cos\theta + b\cos3\theta + c\cos 7\theta$ then find a, b, c, d.
(6 marks)
00
2(b)
Using Newtons Raphson method Solve 3x - $\cos x$ - 1 = 0 Correct to 3 decimal places.
(6 marks)
00
2(c)
Find the stationary points of the function$ x^3 + 3xy^2 - 3x^2 - 3y^2 + 4$ & also find maximum amd minimum values of the function.
(8 marks)
00
3(a)
Show that $ x \hspace{0.1cm}cosec x = 1 + \frac{x^2}{6} + \frac{7}{360}x^4 + ................................$
(6 marks)
00
3(b)
Reduce matrix to PAQ normal form and find 2 non-singular matrices P&Q
$\matrix{1&2&-1&2\\ 2&5&-2&3\\1&2&1&2}$
(6 marks)
00
3(c)
If $y = cos(m \sin^{-1}x)$ prove that $(1- x^2) y_{n+2} - (2n+1)xy_{n+1} + (m^2 - n^2)y_n = 0$
(8 marks)
00
4(a)
State and prove Euler's theorem for three Variables.
(6 marks)
00
4(b)
Show that all the roots of $(x+1)^6 + (x-1)^6 = 0$ are given by $-i\cot \frac{(2k+1)\pi}{12}$ where k = 0, 1, 2, 3, 4, 5
(6 marks)
00
4(c)
Show that the equations
-2x + y + z = a
x -2y + z = b
x + y -2z = c
have no solutions unless a+b+c = 0 in which case they have infinitely many solutions.
Find these solutions when a=1, b=1, c=-2
(8 marks)
00
5(a)
If $z = f(x,y) , x=r\cos\theta$ and $y = r\sin \theta$
$(\frac{\delta z}{\delta x})^2 + (\frac{\delta z}{\delta y})^2 = (\frac{\delta z}{\delta r})^2 + \frac{1}{r^2}(\frac{\delta z}{\delta \theta})^2$
(6 marks)
00
5(b)
If $\cos hx = sec\theta$ prove that
- $x = \log({sec\theta + \tan \theta})$
- $ \theta = \frac{\pi}{2} - 2tan^{-1} (e^{-x})$
(6 marks)
00
5(c)
Solve by Gauss Jacobi Iteration method
5x - y + z = 10
2x + 4y = 12
x + y + 5z = -1
(8 marks)
00
6(a)
prove that $\cos^{-1} [\tan h(\log x)] = \pi - 2(x- \frac{x^3}{3}+\frac{x^5}{5}-----------)$
(6 marks)
00
6(b)
If $y = e^{2x} \sin\frac{x}{2} \cos{x}{2} sin3x$ Find $ y_n$
(6 marks)
00
6(c)
- Evaluate $\lim_{x\rightarrow 0} (\cot x)^{\sin x}$
- Prove that $log[\frac{\sin{x+iy}}{\sin{x-iy}}] = 2i \tan^{-1} (\cot x \tan hy)$
(8 marks)
00